Asymptotic behavior at infinity of solutions of elliptic equations

The prototype problem is the behavior at infinity of all solutions of a linear elliptic equation that have finite L1 norm in a half cylinder 0<y, x in D,D a smoothly bounded domain. We assume that the coefficients of the elliptic operator,as well as the boundary conditions in D, are independent of y. Such a space of solutions can be abstracted as a linear space K of functions f(y), whose values lie in a Banach space B, are translation invariant,|f(y)| integrable,and which are interior compact, an abstract version of ellipticity. We show that the asymptotic behavior as y tends to infinity of such functions is given as a sum of exponential functions contained in K.